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Mathematical Programming Computation

, Volume 11, Issue 1, pp 95–118 | Cite as

Computing feasible points for binary MINLPs with MPECs

  • Lars ScheweEmail author
  • Martin Schmidt
Full Length Paper

Abstract

Nonconvex mixed-binary nonlinear optimization problems frequently appear in practice and are typically extremely hard to solve. In this paper we discuss a class of primal heuristics that are based on a reformulation of the problem as a mathematical program with equilibrium constraints. We then use different regularization schemes for this class of problems and use an iterative solution procedure for solving series of regularized problems. In the case of success, these procedures result in a feasible solution of the original mixed-binary nonlinear problem. Since we rely on local nonlinear programming solvers the resulting method is fast and we further improve its reliability by additional algorithmic techniques. We show the strength of our method by an extensive computational study on 662 MINLPLib2instances, where our methods are able to produce feasible solutions for \({60}{\%}\) of all instances in at most \({10}\,{\hbox {s}}\).

Keywords

Mixed-integer nonlinear optimization MINLP MPEC Complementarity constraints Primal heuristic 

Mathematics Subject Classification

90-08 90C11 90C33 90C59 

Notes

Acknowledgements

This research has been performed as part of the Energie Campus Nürnberg and supported by funding through the “Aufbruch Bayern (Bavaria on the move)” initiative of the state of Bavaria. Both authors acknowledge funding through the DFG Transregio TRR 154, subprojects A05, B07, and B08. Last but not least, we want to express our sincere gratefulness to Stefan Vigerske from GAMS. Without his patient help, the implementations underlying this paper would not have been possible. Thanks a lot, Stefan.

References

  1. 1.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009).  https://doi.org/10.1007/s12532-008-0001-1 MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Baumrucker, B.T., Renfro, J.G., Biegler, L.T.: MPEC problem formulations and solution strategies with chemical engineering applications. Comput. Chem. Eng. 32(12), 2903–2913 (2008).  https://doi.org/10.1016/j.compchemeng.2008.02.010 CrossRefGoogle Scholar
  3. 3.
    Benoist, T., Estellon, B., Gardi, F., Megel, R., Nouioua, K.: Localsolver 1.x: a black-box local-search solver for 0–1 programming. 4OR 9(3), 299 (2011).  https://doi.org/10.1007/s10288-011-0165-9 MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berthold, T.: Heuristic algorithms in global MINLP solvers. Ph.D. thesis, Technische Universität Berlin (2014)Google Scholar
  5. 5.
    Berthold, T., Gleixner, A.M.: Undercover: a primal MINLP heuristic exploring a largest sub-MIP. Math. Program. 144(1), 315–346 (2014).  https://doi.org/10.1007/s10107-013-0635-2 MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Berthold, T., Heinz, S., Pfetsch, M.E., Vigerske, S.: Large neighborhood search beyond MIP. In: Proceedings of the 9th Metaheuristics International Conference (MIC 2011), pp. 51–60 (2011)Google Scholar
  7. 7.
    Berthold, T., Heinz, S., Vigerske, S.: Extending a CIP framework to solve MIQCPs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, pp. 427–444. Springer, New York (2012).  https://doi.org/10.1007/978-1-4614-1927-3_15 CrossRefGoogle Scholar
  8. 8.
    Bonami, P., Cornuéjols, G., Lodi, A., Margot, F.: A feasibility pump for mixed integer nonlinear programs. Math. Program. 119(2), 331–352 (2009).  https://doi.org/10.1007/s10107-008-0212-2 MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bonami, P., Gonçalves, J.P.M.: Heuristics for convex mixed integer nonlinear programs. Comput. Optim. Appl. 51(2), 729–747 (2012).  https://doi.org/10.1007/s10589-010-9350-6 MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    D’Ambrosio, C., Frangioni, A., Liberti, L., Lodi, A.: Experiments with a feasibility pump approach for nonconvex MINLPs. In: Festa, P. (ed.) Experimental Algorithms. Lecture Notes in Computer Science, vol. 6049, pp. 350–360. Springer, Berlin (2010).  https://doi.org/10.1007/978-3-642-13193-6_30 CrossRefGoogle Scholar
  11. 11.
    D’Ambrosio, C., Frangioni, A., Liberti, L., Lodi, A.: A storm of feasibility pumps for nonconvex MINLP. Math. Program. 136(2), 375–402 (2012).  https://doi.org/10.1007/s10107-012-0608-x MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002).  https://doi.org/10.1007/s101070100263 MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Drud, A.S.: CONOPT—a large-scale GRG code. INFORMS J. Comput. 6(2), 207–216 (1994).  https://doi.org/10.1287/ijoc.6.2.207 zbMATHCrossRefGoogle Scholar
  14. 14.
    Drud, A.S.: CONOPT: a system for large scale nonlinear optimization, tutorial for CONOPT subroutine library. Technical Report, ARKI Consulting and Development A/S, Bagsvaerd, Denmark (1995)Google Scholar
  15. 15.
    Drud, A.S.: CONOPT: a system for large scale nonlinear optimization, reference manual for CONOPT subroutine library. Technical Report, ARKI Consulting and Development A/S, Bagsvaerd, Denmark (1996)Google Scholar
  16. 16.
    Fischer, A.: A special Newton-type optimization method. Optimization 24(3–4), 269–284 (1992).  https://doi.org/10.1080/02331939208843795 MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Fischetti, M., Lodi, A.: Local branching. Math. Program. 98(1–3), 23–47 (2003).  https://doi.org/10.1007/s10107-003-0395-5 MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    GAMS Development Corporation: General Algebraic Modeling System (GAMS) Release 24.5.4. Washington, DC, USA (2015). http://www.gams.com. Accessed 10 Aug 2018
  19. 19.
    Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2005).  https://doi.org/10.1137/S0036144504446096 MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gleixner, A., Eifler, L., Gally, T., Gamrath, G., Gemander, P., Gottwald, R.L., Hendel, G., Hojny, C., Koch, T., Miltenberger, M., Müller, B., Pfetsch, M.E., Puchert, C., Rehfeldt, D., Schlösser, F., Serrano, F., Shinano, Y., Viernickel, J.M., Vigerske, S., Weninger, D., Witt, J.T., Witzig, J.: The SCIP Optimization Suite 5.0. Technical Report 17-61, ZIB, Takustr.7, 14195 Berlin (2017)Google Scholar
  21. 21.
    Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137(1), 257–288 (2013).  https://doi.org/10.1007/s10107-011-0488-5 MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Hu, X.M., Ralph, D.: Convergence of a penalty method for mathematical programming with complementarity constraints. J. Optim. Theory Appl. 123(2), 365–390 (2004).  https://doi.org/10.1007/s10957-004-5154-0 MathSciNetCrossRefGoogle Scholar
  23. 23.
    Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010. Math. Program. Comput. 3(2), 103–163 (2011).  https://doi.org/10.1007/s12532-011-0025-9 MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kraemer, K., Kossack, S., Marquardt, W.: An efficient solution method for the MINLP optimization of chemical processes. Comput. Aided Chem. Eng. 24, 105 (2007).  https://doi.org/10.1016/S1570-7946(07)80041-1 CrossRefGoogle Scholar
  25. 25.
    Kraemer, K., Marquardt, W.: Continuous reformulation of MINLP problems. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and Its Applications in Engineering: The 14th Belgian-French-German Conference on Optimization, pp. 83–92. Springer, Berlin (2010).  https://doi.org/10.1007/978-3-642-12598-0_8
  26. 26.
    Liberti, L., Mladenović, N., Nannicini, G.: A recipe for finding good solutions to MINLPs. Math. Program. Comput. 3(4), 349–390 (2011).  https://doi.org/10.1007/s12532-011-0031-y MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)zbMATHCrossRefGoogle Scholar
  28. 28.
    Maher, S.J., Fischer, T., Gally, T., Gamrath, G., Gleixner, A., Gottwald, R.L., Hendel, G., Koch, T., Lübbecke, M.E., Miltenberger, M., Müller, B., Pfetsch, M.E., Puchert, C., Rehfeldt, D., Schenker, S., Schwarz, R., Serrano, F., Shinano, Y., Weninger, D., Witt, J.T., Witzig, J.: The SCIP optimization suite 4.0. Technical Report 17-12, ZIB, Takustr.7, 14195 Berlin (2017)Google Scholar
  29. 29.
    Nannicini, G., Belotti, P.: Rounding-based heuristics for nonconvex MINLPs. Math. Program. Comput. 4(1), 1–31 (2012).  https://doi.org/10.1007/s12532-011-0032-x MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Nannicini, G., Belotti, P., Liberti, L.: A local branching heuristic for MINLPs (2008). arXiv preprint arXiv:0812.2188
  31. 31.
    Rose, D., Schmidt, M., Steinbach, M.C., Willert, B.M.: Computational optimization of gas compressor stations: MINLP models versus continuous reformulations. Math. Methods Oper. Res. 83(3), 409–444 (2016).  https://doi.org/10.1007/s00186-016-0533-5 MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Sahinidis, N.V.: BARON 14.3.1: Global Optimization of Mixed-Integer Nonlinear Programs, User’s Manual (2014)Google Scholar
  33. 33.
    Schmidt, M., Steinbach, M.C., Willert, B.M.: A primal heuristic for nonsmooth mixed integer nonlinear optimization. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization, pp. 295–320. Springer, Berlin (2013).  https://doi.org/10.1007/978-3-642-38189-8_13 CrossRefzbMATHGoogle Scholar
  34. 34.
    Schmidt, M., Steinbach, M.C., Willert, B.M.: An MPEC based heuristic. In: Koch, T., Hiller, B., Pfetsch, M.E., Schewe, L. (eds.) Evaluating Gas Network Capacities, SIAM-MOS series on Optimization, Chapter 9, pp. 163–180. SIAM (2015).  https://doi.org/10.1137/1.9781611973693.ch9
  35. 35.
    Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11(4), 918–936 (2001).  https://doi.org/10.1137/S1052623499361233 MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Stein, O., Oldenburg, J., Marquardt, W.: Continuous reformulations of discrete-continuous optimization problems. Comput. Chem. Eng. 28(10), 1951–1966 (2004).  https://doi.org/10.1016/j.compchemeng.2004.03.011. (Special Issue for Professor Arthur W. Westerberg)CrossRefGoogle Scholar
  37. 37.
    Sun, D., Qi, L.: On NCP-functions. Comput. Optim. Appl. 13(1), 201–220 (1999).  https://doi.org/10.1023/A:1008669226453 MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, Dordrecht (2002)zbMATHCrossRefGoogle Scholar
  39. 39.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99(3), 563–591 (2004).  https://doi.org/10.1007/s10107-003-0467-6 MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005).  https://doi.org/10.1007/s10107-005-0581-8 MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Ugray, Z., Lasdon, L., Plummer, J., Glover, F., Kelly, J., Martí, R.: Scatter search and local NLP solvers: a multistart framework for global optimization. INFORMS J. Comput. 19(3), 328–340 (2007).  https://doi.org/10.1287/ijoc.1060.0175 MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006).  https://doi.org/10.1007/s10107-004-0559-y MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optim. 16(1), 1–31 (2005).  https://doi.org/10.1137/S1052623403426556 MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Wu, B., Ghanem, B.: \(l_p\)-box ADMM: a versatile framework for integer programming. Technical Report (2016). http://arxiv.org/abs/1604.07666. Accessed 10 Aug 2018
  45. 45.
    Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33(1), 9–27 (1995).  https://doi.org/10.1080/02331939508844060 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  1. 1.Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Discrete OptimizationErlangenGermany
  2. 2.Energie Campus NürnbergNurembergGermany

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